| 1. |
- Guo, Lei, et al.
(author)
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Performance Analysis of General Tracking Algorithms
- 1994
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In: Proceedings of the 33rd IEEE Conference on Decision and Control. - Linköping : Linköping University. - 0780319680 ; , s. 2851-2855 vol.3
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Reports (other academic/artistic)abstract
- A general family of tracking algorithms for linear regression models is studied. It includes the familiar LMS (gradient approach), RLS (recursive least squares) and KF (Kalman filter) based estimators. The exact expressions for the quality of the obtained estimates are complicated. Approximate, and easy-to-use, expressions for the covariance matrix of the parameter tracking error are developed. These are applicable over whole time interval, including the transient and the approximation error can be explicitly calculated.
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| 2. |
- Guo, Lei, et al.
(author)
-
Tracking Performance Analysis of the Forgetting Factor RLS Algorithm
- 1992
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In: Proceedings of the 31st IEEE Conference on Decision and Control. - 0780308727 ; , s. 688-693 vol.1
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Conference paper (peer-reviewed)abstract
- The authors present a theoretical analysis for the performance of the standard forgetting factor recursive least squares (RLS) algorithm used in the tracking of time-varying linear regression models. Under some explicit excitation conditions on the regressors, it is shown that the parameter tracking error is on the order O(μ+γ2/μ), where μ=1-λ, λ is the forgetting factor, and γ is the quantity reflecting the speed of parameter variation. Furthermore, for a large class of weakly dependent regressors, simple approximations for the covariance matrix of this error are derived. These approximations are not asymptotic in nature: they hold over all time intervals and for all μ in a certain region.
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| 3. |
- Guo, Lei, et al.
(author)
-
Exponential Stability of General Tracking Algorithms
- 1994
-
Reports (other academic/artistic)abstract
- Tracking and adaptation algorithms are, from a formal point of view, nonlinear systems which depend on stochastic variables in a fairly complicated way. The analysis of such algorithms is thus quite complicated. A first step is to establish the exponential stability of these systems. This is of interest in its own right and a prerequisite for the practical use of the algorithm. It is also a necessary starting point to analyze the performance in terms of tracking and adaptation because that is how close the estimated parameters are to the time-varying true ones. In this paper we establish some general conditions for the exponential stability of a wide and common class of tracking algorithms. This includes least mean squares, recursive least squares, and Kalman filter based adaptation algorithms. We show how stability of an averaged (linear and deterministic) equation and stability of the actual algorithm are linked to each other under weak conditions on the involved stochastic processes. We also give explicit conditions for exponential stability of the most common algorithms. The tracking performance of the algorithms is studied in a companion paper.
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| 4. |
- Guo, Lei, et al.
(author)
-
Exponential Stability of General Tracking Algorithms
- 1995
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In: IEEE Transactions on Automatic Control. - : Institute of Electrical and Electronics Engineers (IEEE). - 0018-9286 .- 1558-2523. ; 40:8, s. 1376-1387
-
Journal article (peer-reviewed)abstract
- Tracking and adaptation algorithms are, from a formal point of view, nonlinear systems which depend on stochastic variables in a fairly complicated way. The analysis of such algorithms is thus quite complicated. A first step is to establish the exponential stability of these systems. This is of interest in its own right and a prerequisite for the practical use of the algorithm. It is also a necessary starting point to analyze the performance in terms of tracking and adaptation because that is how close the estimated parameters are to the time-varying true ones. In this paper we establish some general conditions for the exponential stability of a wide and common class of tracking algorithms. This includes least mean squares, recursive least squares, and Kalman filter based adaptation algorithms. We show how stability of an averaged (linear and deterministic) equation and stability of the actual algorithm are linked to each other under weak conditions on the involved stochastic processes. We also give explicit conditions for exponential stability of the most common algorithms. The tracking performance of the algorithms is studied in a companion paper.
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| 5. |
- Guo, Lei, et al.
(author)
-
Necessary and Sufficient Conditions for Stability of LMS
- 1996
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Reports (other academic/artistic)abstract
- In a recent work (7), some general results on exponential stability of random linear equations are established which can be applied directly to the performance analysis of a wide class of adaptive algorithms including the basic LMS ones without requiring stationarity independency and boundedness assumptions of the system signals The current paper attempts to give a complete characterization of the exponential stability of the LMS algorithms by providing a necessary and sucient condition for such a stability in the case of possibly unbounded nonstationary and non ?mixing signals The results of this paper can be applied to a very large class of signals including those generated fromeg a Gaussian process via a timevarying linear lter As an application several novel and extended results on convergence and tracking performance of LMS are derived under various assumptions Neither stationarity nor Markov chain assumptions are necessarily required in the paper
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| 6. |
- Guo, Lei, et al.
(author)
-
Necessary and Sufficient Conditions for Stability of LMS
- 1995
-
Reports (other academic/artistic)abstract
- Guo and Ljung (1995) established some general results on exponential stability of random linear equations, which can be applied directly to the performance analysis of a wide class of adaptive algorithms, including the basic LMS ones, without requiring stationarity, independency, and boundedness assumptions of the system signals. The current paper attempts to give a complete characterization of the exponential stability of the LMS algorithms by providing a necessary and sufficient condition for such a stability in the case of possibly unbounded, nonstationary, and non-φ-mixing signals. The results of this paper can be applied to a very large class of signals, including those generated from, e.g., a Gaussian process via a time-varying linear filter. As an application, several novel and extended results on convergence and the tracking performance of LMS are derived under various assumptions. Neither stationarity nor Markov-chain assumptions are necessarily required in the paper.
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| 7. |
- Guo, Lei, et al.
(author)
-
Necessary and Sufficient Conditions for Stability of LMS
- 1997
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In: IEEE Transactions on Automatic Control. - : Institute of Electrical and Electronics Engineers (IEEE). - 0018-9286 .- 1558-2523. ; 42:6, s. 761-770
-
Journal article (peer-reviewed)abstract
- Guo and Ljung (1995) established some general results on exponential stability of random linear equations, which can be applied directly to the performance analysis of a wide class of adaptive algorithms, including the basic LMS ones, without requiring stationarity, independency, and boundedness assumptions of the system signals. The current paper attempts to give a complete characterization of the exponential stability of the LMS algorithms by providing a necessary and sufficient condition for such a stability in the case of possibly unbounded, nonstationary, and non-φ-mixing signals. The results of this paper can be applied to a very large class of signals, including those generated from, e.g., a Gaussian process via a time-varying linear filter. As an application, several novel and extended results on convergence and the tracking performance of LMS are derived under various assumptions. Neither stationarity nor Markov-chain assumptions are necessarily required in the paper.
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| 8. |
- Guo, Lei, et al.
(author)
-
Performance Analysis of General Tracking Algorithms
- 1995
-
In: IEEE Transactions on Automatic Control. - : Institute of Electrical and Electronics Engineers (IEEE). - 0018-9286 .- 1558-2523. ; 40:8, s. 1388-1402
-
Journal article (peer-reviewed)abstract
- A general family of tracking algorithms for linear regression models is studied. It includes the familiar least mean square gradient approach, recursive least squares, and Kalman filter based estimators. The exact expressions for the quality of the obtained estimates are complicated. Approximate, and easy-to-use, expressions for the covariance matrix of the parameter tracking error are developed. These are applicable over the whole time interval, including the transient, and the approximation error can be explicitly calculated.
|
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| 9. |
- Guo, Lei, et al.
(author)
-
Performance Analysis of the Forgetting Factor RLS Algorithms
- 1992
-
Reports (other academic/artistic)abstract
- An analysis is given of the performance of the standard forgetting factor recursive least squares (RLS) algorithm when used for tracking time-varying linear regression models. Three basic results are obtained: (1) the ‘P-matrix’ in the algorithm remains bounded if and only if the (time-varying) covariance matrix of the regressors is uniformly non-singular; (2) if so, the parameter tracking error covariance matrix is of the order O(μ + γ2/μ), where μ = 1 - λ, λ is the forgetting factor and γ is a quantity reflecting the speed of the parameter variations; (3) this covariance matrix can be arbitrarily well approximated (for small enough μ) by an expression that is easy to compute.
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| 10. |
- Guo, Lei, et al.
(author)
-
Performance Analysis of the Forgetting Factor RLS Algorithms
- 1993
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In: International journal of adaptive control and signal processing (Print). - : Wiley. - 0890-6327 .- 1099-1115. ; 7:6, s. 525-237
-
Journal article (peer-reviewed)abstract
- An analysis is given of the performance of the standard forgetting factor recursive least squares (RLS) algorithm when used for tracking time-varying linear regression models. Three basic results are obtained: (1) the ‘P-matrix’ in the algorithm remains bounded if and only if the (time-varying) covariance matrix of the regressors is uniformly non-singular; (2) if so, the parameter tracking error covariance matrix is of the order O(μ + γ2/μ), where μ = 1 - λ, λ is the forgetting factor and γ is a quantity reflecting the speed of the parameter variations; (3) this covariance matrix can be arbitrarily well approximated (for small enough μ) by an expression that is easy to compute.
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